Matrix Information Geometry

Abstract

Computation of singular value decomposition (SVD) has been a topic of concern by many numerical linear algebra researchers. Fast SVD has been a very effective tool in computer vision in a number of aspects, such as: face recognition, eye tracking etc. At the present state of the art fast and fixed-point power efficient SVD algorithm needs to be developed for real-time embedded computing. The work in this paper is the genesis of an attempt to build an on-board real-time face and eye tracking system for human drivers to detect loss of attention due to drowsiness or fatigue. A major function of this on-board system is quick customization. This is carried out when a new driver comes in. The face and eye images are recorded while instructing the driver for making specific poses. The eigen faces and eigen eyes are generated at several resolution levels and stored in the on-board computer. The discriminating eigen space of face and eyes are determined and stored in the on-board flash memory for detection and tracking of face and eyes and classification of eyes (open or closed) as well. Therefore, fast SVD of image covariance matrix at various levels of resolution needs to be carried out to generate the eigen database. As a preliminary step, we review the existing symmetric SVD algorithms and evaluate their feasibility for such an application. In this article, we compare the performance of (1) Jacobi's, (2) Hestenes', (3) Golub-Kahan, (4) Tridiagonalization and Symmetric QR iteration and (5) Tridiagonalization and Divide and Conquer algorithms. A case study has been demonstrated as an example.